I am looking at the exercise below in relation to representation theory
We define a group structure on $\lbrace \pm 1, \pm a, \pm b , \pm c \rbrace$ through the relations $a^2 =b^2 =c^2 =abc = 1 $ and denote the resulting group by G.
- Show that $G$ has four inequivalent 1-dimensional complex representations, and find those (Hint: $\lbrace \pm 1 \rbrace $ is a normal subgroup of G and $G / \lbrace \pm 1 \rbrace \simeq \mathbb{Z} / 2 \mathbb{Z} \times \mathbb{Z} / 2 \mathbb{Z} $.)
- Define $\pi: G \to GL_2(\mathbb{C})$ by $\pi (\pm 1 )= \pm I_2 , \pi(\pm a)=\pm \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}, \pi (\pm b)=\pm \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}, \pi(\pm c)= \pm \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix} $. Prove that $\pi$ is a well-defined irreducible representation of $G$.
- Use (1) and (2) to find the character table of G.
I have shown part 2. by using the inner product of the character, but I am having some difficulties in part 1. (and therefore also in part 3). In part 1, I am not quite sure how I am to use a normal subgroup in order to show the wanted. Additionally, I am having some difficulties in deciding what the representations explicitly are.